The Collision Spectrum
The collision transform decomposes the collision invariant into Dirichlet characters. The antisymmetry theorem silences the even half. What remains are the odd characters, each carrying a Fourier coefficient.
This paper opens those coefficients and looks inside.
What the coefficients are
Each coefficient factors. Exactly. Into two pieces that have no business being together.
The first piece is $B_{1,\overline{\chi}}$, the generalized Bernoulli number. This is a finite sum, computable from the character values alone. It has been studied since the 19th century. What makes it relevant here is a classical identity: $|B_1| = (b/\pi)|L(1, \chi)|$. The Bernoulli number IS the L-function at $s = 1$, up to a constant involving $\pi$. It encodes the same information as the class number formula, the same information that governs the distribution of primes in arithmetic progressions.
The second piece is $S_G(\chi)$, a character sum over the diagonal set $G$: the $b$ residue classes modulo $b^2$ whose two base-$b$ digits are equal (00, 11, 22, ..., 99 in base 10). This sum encodes the bin geometry of the digit function. It depends only on the base, not on any prime.
The decomposition theorem:
One factor multiplicative (the L-value, encoding prime distribution). One factor additive (the diagonal sum, encoding digit geometry). Their product is the collision weight at character $\chi$. The digit function's Fourier coefficient is the product of an analytic object and a combinatorial object.
The proof uses three classical ingredients: a formula for the collision invariant in terms of floor-function slices, the Bernoulli identity for character sums over fractional parts, and the vanishing of primitive character sums over cosets. Each ingredient is in the textbooks. The combination, applied to the collision invariant, produces an identity that connects floor-function arithmetic to L-function values.
The moment identity
Once you know the coefficients factor, Parseval's identity on the finite group gives you something for free.
Parseval says: the sum of $|\hat{S}\circ(\chi)|2$ over all characters equals $(1/\phi) \sum |S\circ(a)|2$. The left side, after substituting the decomposition theorem, becomes a sum of $|L(1, \chi)|^2 \cdot |S_G(\chi)|^2$. The right side is a finite sum of squared integers.
The right side is computable. It is a table of integers, squared and summed. No limits, no approximations, no analytic continuation. Just arithmetic.
The left side is a weighted second moment of L-function special values at the edge of the critical strip.
They are equal. The digit function computes an L-value moment exactly.
I want to be clear about what this means. The right side is something you can compute with pencil and paper for small bases. In base 3, there are 6 coprime classes modulo 9, each carrying a centered collision value. Square them, sum them, multiply by $\pi^2 \phi(9) / 9$. You get the weighted second moment of $|L(1, \chi)|^2$ for the primitive odd characters modulo 9. No complex analysis entered the computation. The $\pi$ comes from the Bernoulli-L identity, not from analytic continuation.
At base 5, the diagonal sum satisfies $|S_G(\chi)| = (\sqrt{5}/2)|B_{1,\overline{\chi}}|$ for all 8 primitive odd characters (verified by finite computation, not proved in general). When this holds, the moment identity becomes a fourth moment: $\sum |L(1, \chi)|^4 = c_5 \sum |S\circ(a)|2$. The digit function produces the fourth power of L-values from integer data.
The correlation decay
The decomposition theorem also explains something observed computationally in the companion paper on the collision transform: why the collision coefficients are anti-correlated with the prime character sums.
The coefficient $|\hat{c}(\chi)|^2$ is proportional to $|L(1, \chi)|^2$. The prime sum $P(1, \chi) = \sum \chi(p)/p$ is approximately $\log L(1, \chi)$. So the collision weight grows with $|L(1)|^2$, while the prime sum grows with $|\log L(1)|$. These are not proportional. The logarithm grows more slowly than the square. Characters with large collision weight tend to have large $|L(1)|$ but moderate $|\log L(1)|$. The anti-correlation is built into the factorization.
But the full story is more subtle. The paper analyzes the correction term: the difference between $P(1, \chi)$ and $\log L(1, \chi)$. Through the Apostol-Berndt formula for short character sums, the correction is a sum of twisted L-values whose magnitude is roughly constant ($\approx 1.08$ times $|L(1)|$) but whose phase is uniformly distributed. The uniform phase washes out the tracking at large bases. At small bases, the high variance of the correction allows close tracking. The fidelity between the collision spectrum and individual L-values decays as $c / \log b$.
Where the zeros enter
The analytic continuation of the collision sum into the critical strip carries the decomposition theorem with it. The continued function $\mathcal{F}^\circ(s)$ is a sum of $B_1 \cdot \overline{S_G} \cdot [\log L(s, \chi) - H(s, \chi)]$ over odd characters. Near a zero $\rho$ of $L(s, \chi_0)$, the term $\log L(s, \chi_0)$ diverges. The coefficient of that divergence is $B_1(\overline{\chi_0}) \cdot \overline{S_G(\chi_0)} / \phi$, which involves $L(1, \chi_0)$ through the Bernoulli factor.
The collision sum has a singularity at every L-function zero. The strength of the singularity at depth $s = \rho$ is controlled by the health of the L-function at the edge $s = 1$. The digit function, through the decomposition theorem, sees the zeros. Not directly. Through the weight they carry in the Fourier expansion. But it sees them.
The bridge
This paper is the center of the program. Everything before it (the gate width theorem, the antisymmetry, the convergence) builds the object and its Fourier decomposition. Everything after it (the prime number theorem, the avoidance, the zero-density estimate) uses the decomposition to connect the digit function to the distribution of primes.
The decomposition theorem is the bridge. On one side, the collision invariant: integer-valued, finitely determined, computable from long division. On the other side, L-function special values: analytic, infinite, governing the distribution of primes through the zeros of complex functions. The bridge is exact. Both sides carry the same information, expressed in different languages. The collision weight at a character is the product of an L-value and a geometric factor, and that product equals a Fourier coefficient of the digit function's collision count.
The digit function is the simplest nontrivial arithmetic operation. Its collision invariant is a count of bin coincidences. Its Fourier transform encodes L-values at the edge of the critical strip. The distance between those three sentences is the content of this paper, and closing that distance is what the decomposition theorem does.
Code: github.com/alexspetty/nfield
Paper: The Collision Spectrum
Alexander S. Petty
March 2026
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